You give a cumulative distribution function (CDF), not a probability density function (PDF). Let $F(x)$ be the function you gave and $f(x)$ be its derivative. You can compute
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StirlingDec 10 '12 at 21:10

Correct. Then evaluate the integral $$E[X^{2}] = \int_{0}^{1} x^{2} f(x) dx$$. You can tell that $F(x)$ is a CDF because it does not integrate to 1, while $f(x) = F'(x)$ does.
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StirlingDec 10 '12 at 21:15